Fractal calculus is the simple, constructive, and algorithmic approach to natural processes modeling, which is impossible using smooth differentiable structures and the usual modeling tools such as differential equations. It is the calculus of the future and will have many applications.
This book is the first to introduce fractal calculus and provides a basis for the research and development of this framework. It is suitable for undergraduate and graduate students in mathematics and physics who have mastered general mathematics, quantum physics, and statistical mechanics, as well as researchers dealing with fractal structures in various disciplines.
Author(s): Alireza Khalili Golmankhaneh
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 327
City: Singapore
Contents
Preface
1. Introduction to Analysis of Fractals
1.1 Motivation of fractal analysis
1.2 Fractional calculus
1.3 Fractional space
1.3.1 The Laplace equation in fractional space
1.3.2 Schrödinger’s equation in fractional space
1.3.3 Diffusion equation in fractional space
1.4 Probabilistic approach
1.5 Harmonic analysis approach
1.6 Measure theory approach
1.7 Fα-calculus/fractal calculus approach
2. Basic Tools
2.1 Real analysis
2.2 Measure theory
3. Fractal Cantor Like Sets
3.1 Cantor ternary set
3.2 The Devil’s staircase function
3.3 The Smith-Volterra-Cantor sets
3.4 The fat Cantor sets
3.5 The middle-β Cantor sets
3.6 Generalized Smith-Volterra-Cantor sets
4. Fractal Calculus
4.1 The mass function of fractal sets
4.1.1 The mass function under translation and scaling
4.2 The integral staircase function
4.3 The γ-dimension of fractal sets
4.4 The α-perfect sets
4.5 Fα-integration on fractal sets
4.5.1 A few properties of Fα-integral on fractal sets
4.6 Differences and similarities between Fα-integral and Riemann-Stieltjes integral
4.7 Fα-differentiation on fractal sets
4.7.1 The linearity of the fractal derivatives
4.8 Fundamental theorem of Fα-calculus of fractal sets
4.9 Function spaces in Fα-calculus on fractal sets
4.9.1 Spaces of Fα-differentiable functions on fractal sets
4.9.2 Spaces of Fα-integrable functions on fractal sets
4.10 Analogues of abstract Sobolev spaces on fractal sets
4.11 Conjugacy between fractal calculus and ordinary calculus
4.11.1 Review of the Riemann integral on R
4.11.2 Conjugacy between fractal integral on fractal sets and the ordinary integrals
4.11.3 Conjugacy between fractal derivative on fractal sets and the ordinary derivative
4.11.4 Conjugacy between the Sobolev spaces on fractal sets and ordinary Sobolev spaces
4.12 Calculus on fractal curves in Rn
4.12.1 Parameterizing fractal curves
4.12.2 Properties of Fα-calculus on fractal curves
4.12.3 Fundamental theorem of Fα-calculus on fractal curves
4.13 Conjugacy of Fα-calculus on fractal curves and ordinary calculus
4.14 Function space on fractal curves
4.14.1 Spaces of Fα-differentiable functions on fractal curves
4.14.2 Fα-integrable functions on fractal curves
4.15 Analogues of abstract Sobolev spaces on fractal curves
5. Local Fractal Differential Equations
5.1 Local differential equations with α-order on fractal sets and curves
5.2 The local Fourier transforms on fractal sets
5.2.1 Fourier transform on fractal curves
5.2.2 Solving fractal differential equation by fractal Fourier transform
5.3 The fractal Laplace transform
5.3.1 Solving fractal differential equation by fractal Laplace transform
5.4 Sumudu transform on fractal set
5.4.1 Solving the fractal differential equation via the fractal Sumudu transform
6. Stability of Fractal Differential Equations
6.1 Lyapunov stability of fractal differential equations
6.2 Hyers-Ulam stability of the first order equation
6.2.1 Hyers-Ulam stability α-order fractal differential equation
6.3 Existence and uniqueness theorems for α-order fractal linear differential equations
6.4 The Lie method on fractal calculus
6.4.1 Symmetry condition of fractal differential equations
7. Generalization of Fractal Calculus
7.1 Non-local fractal calculus
7.2 Scale change on the local and non-local fractal derivatives
7.3 Gauge integral on fractal calculus
7.3.1 *Fα-differentiable functions
7.4 Random variables and processes on fractal
7.4.1 Random variables on fractal
7.4.2 Random processes on fractals
7.5 Mean square fractal calculus
7.5.1 Stochastic differential equation on fractal sets
7.6 Hilbert spaces on fractals
7.7 Self-adjoint fractal differential operator
7.8 Strum-Liouville equation on fractals
7.8.1 Fractal Lengendre’s equation
7.8.2 Fractal Legendre polynomials
7.8.3 Fractal Rodrigues’ formula
7.8.4 Fractal Hermite’s equation
7.8.5 Fractal Hermite’s polynomials
7.8.6 Fractal Rodrigues’ formula
7.9 Scale transform of the fractal Hermite and the Legendre polynomials
7.10 Fractal finite difference and fractal derivative
7.11 Fractal difference and its relation with fractal differential equations
7.12 Numerical method for solving fractal differential equation
7.13 Laplace equations on fractal cubes
7.14 Laplace equations on fractals
7.15 Fractal calculus on Cantor tartan spaces
7.16 New measure based on the staircase function
8. Applications of Fractal Calculus
8.1 Motion in Fractally distributed medium
8.2 Relaxation in glassy materials
8.3 A fractal time diffusion equation
8.4 Fokker-Plank equation on fractal curves
8.5 Random walk on fractal curves
8.5.1 Discrete random walk on fractal curves
8.5.2 Random walks with Non-Gaussian distributions on fractals
8.5.3 First passage time on fractal curves
8.6 Langevin equation on fractal curves
8.6.1 Model of noise
8.6.2 Solution of the Langevin equation
8.7 Sub- and super-diffusion on fractal
8.8 Fractal over-damped Langevin equation
8.9 Fractal under-damped Langevin equation
8.10 Fractal scaled Brownian motion
8.11 Diffraction fringes from fractal sets
8.12 Schrödinger equation on fractal
8.13 Partition function of systems on fractal
8.14 Density of states on fractal spaces
8.15 Newton’s second law on fractal time
8.16 Series resistor, an inductor, and a capacitor involving fractal time
8.16.1 Series RLC circuit involving fractal time
8.16.2 Parallel RLC circuit involving fractal time
8.17 Kepler’s third law on fractal time
8.18 Lagrangian and Hamiltonian mechanics on fractal
8.18.1 Poisson bracket on fractals
8.19 Gradient, divergent, curl and Laplacian on fractal curves
8.20 Fractal differential forms
8.21 Maxwell’s equations on fractals
8.22 Anomalous diffusion on fractal Cantor tartan
8.23 Brownian motion on fractal sets
8.24 Fractional Brownian motion on fractal sets
8.24.1 First representation
8.24.2 Second representation
8.24.3 Third representation
8.25 Spectral density of fractional Brownian motion on fractal sets
8.26 Fractal logistic equation
8.27 Distribution on fractals
8.28 Hierarchy of stable distributions on fractals
8.29 Kronig-Penney model on fractals
8.30 Tsallis entropy on fractals
8.31 Fractal Schwarzschild metric
8.32 Fractal damped oscillators on Finsler manifold
8.33 Fractal electromagnetic fields in Randers spaces
8.34 The Boltzmann/Vlasov-Boltzmann on fractal time
8.35 Fluids equation on fractal time
8.36 Fractal free unmagnetized electrons
8.37 Fractal wave in unmagnetized plasma
8.38 Casimir effect on fractal spaces
8.39 Fractal models for the viscoelasticity
Appendix A Appendix
A.1 Algorithm for the mass function on fractal sets
A.1.1 The steps of algorithms
A.2 Algorithm for the mass function on fractal curves
A.3 Algorithm for α-dimension of fractal curves
A.4 Comparison between γ-dimension and box dimension
A.5 Comparison between γ-dimension and Hausdorff dimension
A.6 Fα-integrating of a function
A.7 Repeated Fα-derivative
A.8 Repeated Fα-integration
A.9 A few analogies between Fα-calculus and ordinary calculus
Bibliography
Index
